I find QM quite confusing, in that one can observe only the eigenvalues and not the state itself. Why is it specifically, or is this wrong conceptualization?
Also, how does particle-ness relate to the eigenvalues?
Eigenvalues come from linear algebra. I think a difficult think in general with understanding them is often the failure of most middle/high school math teachers to teach matrix operations at all. (I’m guessing because matrix multiplication never shows up on SAT/ACT). Here’s a good explanation for the math on finding eigenvalues and eigenvectors.
But basically eigenvalues are going to be associated with certain matrixes/vectors. You take a “Hamiltonian” of a system, which is a way of describing possible energy values in the system, and it’ll give you a set of possible answers - pairs of eigenvalues and eigenvectors that describe the system.
In effect - you get things like the quantum numbers. That the 1st energy level has 1 subshell can hold 2 electrons, both with opposing spins. That the 2nd energy level has a 2s subshell that holds two, that 2p holds six. You get your n (1st energy level, 2nd so on as you go down periods of the periodic table), l (subshell - don’t get a SPeeDy F), m (which breaks down where in the subshell they are) and the need for opposing spins.
Thank you for in-depth explanation! Though I already know the eigenvalues and eigenvectors, as a math major. What I am curious of is: why can’t we only observe e.g. energy values? I heard that one can only observe commutative operators or something, but honestly why is quite unclear.
I’ll try to dig out Griffith for a better explanation but has to do with the fact that when you do a partial derivative you kinda lose information I guess?
(Idk, this is heady trying to make math into reality shit and I got a “c” in the class (for reasons partially related to other things) - also, there might be a way to do latex in markdown but I’m a bit too stoned to figure out, look up Schrödinger equation on wiki for maybe a helpful visual aid)
So go back how often we do implicit differential because it’s just an opportunity to look at how sexy the chain rule is. d(xy)/dx = xy’+x’y god fucking dammit that gorgeous
But okay. Think about position and velocity. Velocity is the derivative of position right (and also connected to energy - KE = 1/2mv^2 and E = mc^2 lol)
But since velocity is a derivative of position, it loses information. d(mx+b)/dx turns into m, no way to ever get b back with an initial value condition.
Then - omigod, when you take a partial - you have to ignore dependence. curlyd(xy+by)/curlydx turns into y and then things is really fucked if there was any dependence on y (ie, doing curlyd(xy+by)/curlydy would give you a different answer if you did that first order matters I guess)
There are some operators that are just exclusionary. Once you chose to look for one, you’ve discounted the chance of finding the other. Taking position versus taking energy/velocity. And then the fucky thing there is lots of shits mass is measured in eV/c^2
(I’m neglecting a proper discussion of momentum which is 100% where someone can come in and humiliate me. Please do so.)
I find QM quite confusing, in that one can observe only the eigenvalues and not the state itself. Why is it specifically, or is this wrong conceptualization? Also, how does particle-ness relate to the eigenvalues?
Eigenvalues come from linear algebra. I think a difficult think in general with understanding them is often the failure of most middle/high school math teachers to teach matrix operations at all. (I’m guessing because matrix multiplication never shows up on SAT/ACT). Here’s a good explanation for the math on finding eigenvalues and eigenvectors.
But basically eigenvalues are going to be associated with certain matrixes/vectors. You take a “Hamiltonian” of a system, which is a way of describing possible energy values in the system, and it’ll give you a set of possible answers - pairs of eigenvalues and eigenvectors that describe the system.
In effect - you get things like the quantum numbers. That the 1st energy level has 1 subshell can hold 2 electrons, both with opposing spins. That the 2nd energy level has a 2s subshell that holds two, that 2p holds six. You get your n (1st energy level, 2nd so on as you go down periods of the periodic table), l (subshell - don’t get a SPeeDy F), m (which breaks down where in the subshell they are) and the need for opposing spins.
Thank you for in-depth explanation! Though I already know the eigenvalues and eigenvectors, as a math major. What I am curious of is: why can’t we only observe e.g. energy values? I heard that one can only observe commutative operators or something, but honestly why is quite unclear.
I’ll try to dig out Griffith for a better explanation but has to do with the fact that when you do a partial derivative you kinda lose information I guess?
(Idk, this is heady trying to make math into reality shit and I got a “c” in the class (for reasons partially related to other things) - also, there might be a way to do latex in markdown but I’m a bit too stoned to figure out, look up Schrödinger equation on wiki for maybe a helpful visual aid)
So go back how often we do implicit differential because it’s just an opportunity to look at how sexy the chain rule is. d(xy)/dx = xy’+x’y god fucking dammit that gorgeous
But okay. Think about position and velocity. Velocity is the derivative of position right (and also connected to energy - KE = 1/2mv^2 and E = mc^2 lol)
But since velocity is a derivative of position, it loses information. d(mx+b)/dx turns into m, no way to ever get b back with an initial value condition.
Then - omigod, when you take a partial - you have to ignore dependence. curlyd(xy+by)/curlydx turns into y and then things is really fucked if there was any dependence on y (ie, doing curlyd(xy+by)/curlydy would give you a different answer if you did that first order matters I guess)
There are some operators that are just exclusionary. Once you chose to look for one, you’ve discounted the chance of finding the other. Taking position versus taking energy/velocity. And then the fucky thing there is lots of shits mass is measured in eV/c^2
(I’m neglecting a proper discussion of momentum which is 100% where someone can come in and humiliate me. Please do so.)